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If N Is Odd Positive Integer And 1 X X 2 X 3 N Sigma R 0 To 3n Ar Xr Then A0 A1 A2 A3 A3n Is Equal To

If n is odd positive integer and (1+x+x^2+x^3)^n = sigma r = 0 to 3n arx^r, then a0-a1+a2-a3+...-a3n is equal to (1) 4^n (2) 1 (3) -1 (4) 0... View Article

The value of 1+(1/3)+(1.3)/(3.6) + (1.3.5)/(3.6.9)+(1.3.5.7)/(3.6.9.12) +…. is equal to

(1) (2) (3) (4) Solution: 1+(⅓)+(1.3)/(3.6) + (1.3.5)/(3.6.9)+ (1.3.5.7)/(3.6.9.12) +... Let y = 1+(⅓)+(1.3)/(3.6) +... View Article

Let The Positive Numbers A B C D Be In Ap Then Abc Abd Acd Bcd Are

Solution: Given a,b,c,d are in AP. So d,c,b,a are in AP. 1/d, 1/c, 1/b, 1/a are in HP. So abcd/d, abcd/c, abcd/b,abcd/a are in HP =>... View Article

If Sn Sigma K 1 To 4n 1 K K 1 2k 2 Then Sn Can Take Values

Solution: ∑k=14n (-1)k(k+1)/2k2 = -12-22+32+42-52-62+72+82-....+(4n)2 = (-12+32-52+72-...+(4n-1)2)+(-22+42-62+82-..+(4n2) =... View Article

Let Alpha Beta Be The Roots Of X2 X P 0 And Gamma And Delta Be Roots Of X2 4x Q 0 If Alpha Beta Gamma Delta Are In Gp Then Integral Values Of P And Q Respectively Are

Solution: Given α, β are the roots of x2-x+p = 0 Sum of roots, α+β = 1…(i) Product of roots, αβ = p γ, δ are the roots of x2-4x+q = 0 Sum... View Article

Sigma K 1 To 2n 1 1 K 1 K2 Equals

Solution: ∑k=12n+1(-1)k-1k2= 12-22+32-42+...(2n)2+(2n+1)2 = [12+32+52+...(2n+1)2]-[22+42+62+...+(2n)2] =... View Article

Let Alpha Beta Denote The Cube Roots Of Unity Other Than 1 And Alpha Not Equal Beta If S Sigma Then The Value Of S Is

Solution: S = ∑n=0302 (-1)n(α/ β)n Cube roots of unity are 1, ω, ω2. Take β = ω and α = ω2 ∑n=0302(-1)n(α/ β)n = ∑n=0302(-1)n( ω2/ω)n... View Article

The Sum Of The Series Sigma N 8 To 17 1 N 2 N 3 Is Equal To

Solution: S = ∑n=817 1/(n+2)(n+3) = ∑n=817 1/(n+2) -1/(n+3) = ((1/10)-(1/11)+(1/11)-(1/12)+...+(1/19)-(1/20)) = (1/10)-(1/20) = 1/20... View Article

The Value Of 2 Power 1 By 4 4 Power 1 By 8 Infinity Is

Solution: 21/4 × 41/8 ×81/16 ... ∞ S = (1/4)+(2/8)+(3/16)+... Divide by 2 S/2 = (1/8)+(2/16)+(3/32)+... Subtracting S/2 =... View Article

The Sum Of The Series 1 3x 6×2 10×3 Infinity Will Be

Solution: Let S = 1+3x+6x2+10x3 + ...∞ ..(i) Multiply by x xS = x+3x2+6x3+10x4+ ...∞ (ii) (i)-(ii) (1-x)S = 1+2x+3x2+4x3+....∞ (iii)... View Article

If Y 3x 1 3 X 1 X Real Then The Least Value Of Y Is

Solution: Given y = 3x-1 + 3- x-1 = (3x/3)+(1/3x×3) We know AM ≥GM ((3x/3)+(3/3x))/2 ≥ √(3x/3)(1/3×3x) ((3x/3)+(3/3x))/2 ≥ √(1/9)... View Article

Let A B C Be In Ap And A 1 B 1 C 1 If X 1 A A2 Infinity Y 1 Then X Y And Z Are In

Solution: Since a,b,c in AP 2b = a+c x = 1 + a + a2 + ...+ ∞ This is an infinite GP. Sum = 1/(1-r) x = 1/(1-a) ..(i) y = 1 + b + b2 +... View Article

If Fx Cos 2x Plus Sec 2x Then

Solution: We know AM ≥ GM (cos2x+sec2x )/2 ≥ √(cos2x sec2x) (cos2x+sec2x ) ≥ 2√(cos2x (1/cos2x) f(x) ≥ 2 Hence option (4) is the answer.... View Article

If A1 A2 G1 G2 And H1 H2 Are Two Am Gm And Hm Between Two Quantities Then The Value Of

If A1, A2, G1 , G2 and H1, H2 are two AM's, GM's and HM's between two quantities, then the value of G_1G_2/H_1H_2 is (1) (A_1+A_2)/H_1+H_2 (2)... View Article

If H1 And H2 Are Two Harmonic Means Between Two Positive Numbers A And B A Not Equal B A And G Are The Arithmetic And Geometric Means Between A And B Then H2 Plus H1 By H1h2 Is

If H1 and H2 are two harmonic means between two positive numbers a and b (a not equal b), A and G are the arithmetic and geometric means between... View Article

If Am Of Two Numbers Is Twice Of Their Gm Then The Ratio Of Greatest Number To Smallest Number Is

Solution: Let a and b be the two numbers. AM = 2 GM (a+b)/2 = 2√(ab) (a+b) = 4√(ab) Squaring both sides (a+b)2 = 16ab ..(i) a2+b2-14ab =... View Article

Three Numbers Whose Sum Is 15 Are In Ap If They Are Added By 1 4 And 19 Respectively Then They Are In Gp The Numbers Are

Solution: Let a-d,a,a+d are the three numbers in AP. Given sum = 15 a-d+a+a+d = 15 3a = 15 a = 15/3 = 5 When added 1,4,19 respectively... View Article

If A Is Positive And Also A G Are The Arithmetic Mean And The Geometric Mean Of The Roots Of X2 2ax A2 0 Respectively Then

Solution: Given x2-2ax+a2 = 0 (x-a)2 = 0 x-a = 0 => x = a Two roots are a. AM of roots, A = (a+a)/2 = a GM of roots, G = √a2 = a So... View Article

What Is The Inflorescence Of Cotton

The inflorescence of cotton is scorpioid cyme. It is a type of monochasial cyme, where lateral branches alternate on both sides of the axis.... View Article

What Is The Inflorescence Of Marigold

The inflorescence of a marigold is capitulum or head. It is a type of racemose inflorescence. The peduncle is short and flattened. Flowers are... View Article